I am trying to prove that lim N → ∞ 1 2 N ∑ k = 1 N [ cos ⁡ ( ( x − k L ) ⋅ q ) + cos ⁡ ( ( x + k L ) ⋅ q ) ] = 0for every x , L ∈ R + with x ≤ L and for every q ≠ 2 n π L , n ∈ Z .

Question

I am trying to prove that      lim          N      →      ∞            1          2      N            ∑          k      =      1        N    [  cos  ⁡  (  (  x  −  k  L  )  ⋅  q  )  +  cos  ⁡  (  (  x  +  k  L  )  ⋅  q  )  ]  =  0for every   x  ,  L  ∈            R        +   with   x  ≤  L and for every   q  ≠            2      n      π        L  ,   n  ∈      Z  .

America Barrera · Accepted Answer

cos  ⁡  (  α  ±  β  )  =  cos  ⁡  α  cos  ⁡  β  ∓  sin  ⁡  α  sin  ⁡  β  . Plugging this into your expression yields   cos  ⁡  (  x  q  )      1    N        ∑          k      =      1        N    cos  ⁡  (  k  L  q  )  . Also,   sin  ⁡  (  α  ±  β  )  =  sin  ⁡  α  cos  ⁡  β  ±  sin  ⁡  β  cos  ⁡  α  . Therefore,   2  sin  ⁡  (      α    2    )  cos  ⁡  (  k  α  )  =  sin  ⁡  (  (  k  +      1    2    )  α  )  −  sin  ⁡  (  (  k  −      1    2    )  α  )  . This allows for a telescopic sum argument.       1    N        ∑          k      =      1        N    cos  ⁡  (  k  L  q  )  =            sin      ⁡      (      (      N      +              1        2            )      L      q      )      −      sin      ⁡      (              1        2            L      q      )              2      N      sin      ⁡      (                        L          q                2            )            →          N      →      ∞        0

I am trying to prove that <munder> <mo movablelimits="true" form="prefix">lim <mrow clas

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